Computing Eigenvalues of Large Scale Hankel Tensors

Abstract

Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of $m$th order $n$ dimensional Hankel tensors, where $n$ is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about $\mathcal{O}(mn\log (mn))$. Using the Cayley transform, we obtain an effective curvilinear search scheme. Then, we show that every limiting point of iterates generated by the new algorithm is an eigen-pair of Hankel tensors. Without the assumption of a second-order sufficient condition, we analyze the linear convergence rate of iterate sequence by the Kurdyka-Łojasiewicz property. Finally, numerical experiments for Hankel tensors, whose dimension may up to one million, are reported to show the efficiency of the proposed curvilinear search method.

References in corpus (3)

• DFacTo: Distributed Factorization of Tensors
• Positive Semi-Definiteness and Sum-of-Squares Property of Fourth Order Four Dimensional Hankel Tensors
• Computing tensor eigenvalues via homotopy methods

Cited by in corpus (3)

• Inheritance Properties and Sum-of-Squares Decomposition of Hankel Tensors: Theory and Algorithms
• The positive semi-definite cone and sum-of-squares cone of Hankel form
• Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph